I want to identify the least natural number $n$ (of course, it suffices to solve this problem for the reals, and then take the floor) such that

$$-c \text{Ei}\left(-e^{\frac{a-d}{c}} (n+1)\right)+a-b (n+1)+c \log (n+1)+\gamma c < 0,$$

where $\text{Ei}$ is the exponential integral, $a,b, c, d$ are arbitrary real constants, and $\gamma$ is the Euler-Mascheroni constant.

I have tried moving the terms around etc., to no avail; Mathematica also does not seem able to solve this.

I was wondering if there are any easy ways to solve this, or at least simplify it?


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